## Question

###### 276Testing_and Confidence Regions Chizar [(jii) Consider the model 0Falv)0 # 0 "1 Foly), 0 = 0.To see whether the new treatment is beneficial, we test H : 0 < versus k 0 > U. Assume that Fo has density foly) . Show that the UMP test is based on the slatistic 5RLI Fo(Y) 9/Let Xi beiid with distribution function Flr) . We want test whether F is exponential, F(c) = 1 exp(~r)r > 0, or Weibull, F(r) = 0 > 0. Find the MP test for testing H : 0 exp(-I" ) not UMP versus K 01 > S

276 Testing_and Confidence Regions Chizar [ (jii) Consider the model 0Falv) 0 # 0 "1 Foly), 0 = 0. To see whether the new treatment is beneficial, we test H : 0 < versus k 0 > U. Assume that Fo has density foly) . Show that the UMP test is based on the slatistic 5RLI Fo(Y) 9/Let Xi beiid with distribution function Flr) . We want test whether F is exponential, F(c) = 1 exp(~r)r > 0, or Weibull, F(r) = 0 > 0. Find the MP test for testing H : 0 exp(-I" ) not UMP versus K 01 > Show that the lestis Show that under the assumptions of ncotet complete _ 4.3.2 the class of all Bayes tests Hint: Consider the class of all Bayes tests of H "{00 } x{01 } varies between Qw versus K when' and Show that under the assumptions of Teonete H : Oo versus K 01 is of the fori L3Land 0-1 loss, every Bayes test for for some Kint: Bayes lest rejects (accepts) H if p(z, O)da(0) / P(z, O)d(0) (<) 1. The left-hand side equals L(T; Oo )dn(0) L(e, Vo )dg(0) The numerator an increasing funclion of T(z). the denominator decreasing- 12 . Show that under the assumptions of Theorem 4..1; IUMP 0o versus K esliqg H : 0 2 Froblems for Section 4.4 Let sample from oma population with uunknor Glu;t